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Suppose the amplitude of a forced oscillator is given as $x_m=\frac{F_m}{\left[m^2\right({\omega }_d^2-{\omega }^2)+b^2{\omega }_d^2{]}^{\frac{1}{2}}}$ where Fm is the (constant) amplitude of the external oscillating force. At resonance,

what are the amplitude and velocity amplitude of the oscillating object?

For a particle moving along $x$ - axis, the acceleration $a$ of the particle in terms of its $x$- coordinate $x$ is given by $a=-9x$, where $x$ is in meters and $a$ is in ${\text{m/s}}^2$. Take acceleration, velocity and displacement in positive $x$ - direction as positive. The initial velocity of particle at $x=0$ is $u=+6\text{m/s}$. The velocity of particle at $x=2\text{m}$ will be

In the figure given below, there are two blocks A and B of weight $20N$ and $100N$ respectively. These are being pressed against a wall by an external force $F$ as shown in figure. If the coefficient of static friction between the blocks is $0.1$ and between block B and the wall is $0.15$, then the frictional force applied by the wall on block B will be

The approximate depth of an ocean is $2700\text{m}$. The compressibility of water is $45.4\times {10}^{-11}{\text{Pa}}^{-1}$ and density of water is ${10}^3{\text{kg/m}}^3$. Fractional compression of water will be:

[Take $g=10{\text{m/s}}^2$]

A student finds that the screw gauge that he is using to measure the thickness of a metal sheet has 50 divisions on the circular scale. If the pitch of the screw gauge is 0.05 cm, what is its least count?

A time dependent force $F=\beta t$ acts on a particle of mass $4\text{kg}$. If the particle starts from rest, the work done by the force during the first $1$ second will be [$\beta$ is a (+)ve constant]

8. A solid homogeneous sphere is moving on a rough horizontal surface, partly rolling and partly sliding During this kind of motion of the sphere:

(a) total kinetic energy is conserved

(b angular momentum of the sphere about the point of contact with the plane is conserved (c) only the rotational kinetic energy about centre of mass is conserved

(d) angular momentum about centre of mass is conserved

The total number of points of local maxima and local minima of the function $f\left(x\right)=\{\begin{array}{c}(2+x{)}^3,-3<x\le -1\\ x^{\frac{2}{3}},-1<x<2\end{array}$ is

An open knife edge of mass $m$ is dropped from a height $h$ on a wooden floor. If the blade penetrates up to the depth $d$ into the wood, the average resistance offered by the wood to the knife edge is

A stone of mass $1\text{kg}$ tied to a light string of length $\frac{10}{3}\text{m}$ is whirled in a vertical circle. If the ratio of the maximum tension to minimum tension is 4 and $g=10{\text{m/s}}^2$, then the speed of the stone at the highest point of the circle is

Collision frequency of a nitrogen molecule in a cylinder containing nitrogen at $2\text{atm}$ and temperature of ${17}^{\circ }\text{C}$

(Assume radius of a nitrogen molecule to be $1.0\stackrel{\circ }{\text{A}}$)

A car is moving on a levelled circular road of radius of curvature $300\text{m}$. If the coefficient of static friction is $0.3$ and acceleration due to gravity is $10m{s}^{-2}$, then maximum allowable speed for the car will be (in $km{h}^{-1})$

The magnetic field $\left(B\right)$ due to revolving electron at the center is proportional to $\frac{Z^x}{n^5}$. The value of $x$ is -

The magnetic field due to a short magnet, at a point on its axis, at a distance $X\text{cm}$ from the middle point of the magnet, is $200\text{Gauss}$. The magnetic field at a point on the equatorial axis, at a distance $X\text{cm}$ from the centre of magnet, will be

The magnetic field insides a toroid of radius $r$ is $B$. If the radius of toroid is quadrupled and number of turns is increased by $8$ times keeping the current constant. Find the new magnetic field inside the core of toroid.

If a matrix $A=\left[a_{ij}\right]$ is given as $A=\left[\begin{array}{cccc}1& 2& 3& -1\\ 3& -2& 1& 0\\ 0& 3& 2& 4\end{array}\right],$ then the value of $\sum _{i=1}^3{a}_{ii}=$

In an electron microscope, electrons accelerated by a voltage of $50\text{kV}$ are used. In an optical microscope of the same aperture, yellow light of wavelength $5900\mathring{\text{A}}$ is used. The resolving power of the electron microscope will be $n$ times higher than that of the optical microscope, where $n$ is of the order of

Rain is falling vertically with a speed of $12\sqrt{3}m/s$. A woman rides a bicycle with a speed of $12m/s$ from East to West direction. What is the velocity of rain relative to the woman?

(Take east and vertically upward as positive x-axis and positive y-axis respectively.)

Maximum and minimum magnitudes of the resultant of two vectors of magnitudes $P$ and $Q$ are found to be in the ratio of $3:1$. Then

2 kg of ice at - ${20}^{\circ }C$ is mixed with 5 kg of water at ${20}^{\circ }C$ in an insulating vessel having a negligible heat capacity. Calculate the final mass of water remaining in the container. It is given that the specific heats of water and ice are 1 kcal/kg per ${}^{\circ }C$ and 0.5 kcal/kg/${}^{\circ }C$ while the latent heat of fusion of ice is 80 kcal/kg

A metallic block carrying current $i$ is subjected to a uniform magnetic field $\overrightarrow{B}$ as shown in figure. The moving charges in the metal experience a force $F$ given by _______ , which results in the lowering of the electric potential of the face_______. Assume the speed of the charged particle to be $v$, then select the option to fill in the blanks.

A small spherical droplet of density $d$ is floating exactly half immersed in a liquid of density $\rho$ and surface tension $T$. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet):

Two uniform rods $\text{A}$ and $\text{B}$ are rigidly joined end to end and is pivoted at point $P$, such that it can freely rotate about point $P$ in a vertical plane. A small object moving horizontally, hits the lower end of the combination and stick to it. What should be the velocity of the object so that the system could just be raised to the horizontal position ?

Two charges $+3.2\times {10}^{-19}Cand-3.2\times {10}^{-19}C$ placed $2.4A^o$ apart form an electric dipole. It is placed in a uniform electric field of intensity $4\times {10}^5{Vm}^{-1}$ , the workdone to rotate the electric dipole from the equilibrium position by ${180}^o$ is ......$\times {10}^{-23}J$

A particle is executing simple harmonic motion. Its displacement is given by $x=5\sin \pi t$ where $x$ is in $\text{cm}$ and $t$ in seconds. How long will the particle take to move from the position of equilibrium to the position of maximum displacement ?

What is the term given to the block placed on the left side?

Two masses, nm and m, start simultaneously from the intersection of two straight lines with velocities v and nv respectively. It is observed that the path of their centre of mass is a straight line bisecting the angle between the given straight lines. Find the magnitude of the velocity of centre of mass.[Here $\theta$ = angle between the lines]

A parallel plate capacitor is constructed using three different dielectric materials, as shown in the figure. Find the effective capacitance across $A$ and $B$.

Two small balls A and B, each of mass m, are joined rigidly to the ends of a light rod of length L (figure 10-E10). The system translates on a frictionless horizontal surface with a velocity $v_0$ in a direction perpendicular to the rod. A particle P of mass m kept at rest on the surface sticks to the ball A as the ball collides with it. Find (a) the linear speeds of the balls A and B after the collision, (b) the velocity of the centre of mass C of the system A + B + P and (c) the angular speed of the system about C after the collision.